Ideal gas laws
Ideal Gas Equation Example 1 Figuring out the number of moles of gas we have using the ideal gas equation: PV=nRT.
Ideal Gas Equation Example 1
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- In the last video we hopefully learned the intuition behind
- the ideal gas equation, that pressure times volume is equal
- to the number of molecules we have times some constant times
- the temperature.
- And that's all nice and it hopefully it makes sense to
- you how all of these fit together.
- That pressure should be inverse to volume and that's
- why you're multiplying both sides by each other.
- You could take volume and put it on
- this side of the equation.
- Or that pressure should be proportional to the number of
- particles and the temperature.
- But now let's apply it and actually do some problems.
- Because just knowing this isn't good enough.
- So let's say that I have a two liter container, or let's say
- a two liter balloon, containing hydrogen gas.
- And that's hydrogen as a diatomic molecule.
- So each molecule has two hydrogens in it.
- And let's say I'm measuring it at 30 degrees Celsius.
- My brain is really malfunctioning.
- And let's say that the pressure on the outside of the
- balloon, we've measured at two atmospheres.
- So my question to you is how many moles of
- hydrogen do we have?
- So let's apply our ideal gas equation.
- And since we're dealing with liters and atmospheres, we
- have to make sure we use the right
- proportionality constant.
- So our pressure is given in atmospheres.
- Let me write down all the units, actually.
- So we have 2 atmospheres times our volume is 2 liters, is
- equal to n.
- n is the number of particles we care about, and we care
- about it in moles, but let's just write n there for now.
- Is equal to n times R.
- I'll do R in a second.
- Times T.
- Now you might be tempted to just put 30 degrees in there.
- But in all of these problems-- in fact in general, whenever
- you're doing any of these gas problems or thermodynamics
- problems, or any time you're doing math with temperature--
- you should always convert into Kelvin.
- And just as a bit of review as to what Kelvin is, it's just a
- different scale.
- So for example, the lowest possible temperature that that
- can be achieved in the universe, when you think about
- it in Celsius, let me draw a little temperature scale here.
- So if that's the temperature scale.
- I'll draw two, one for Celsius and one for Kelvin.
- So the lowest possible temperature that can be
- achieved in the universe, and when we say the lowest
- possible temperature that means that the average kinetic
- energy of the molecules or the atoms are zero.
- They're just not moving.
- They're just stationary.
- So in Celsius, it's minus 273.15 degrees Celsius.
- So zero might be some place over here.
- Zero, that's where water freezes.
- And then 100 degrees, that's where water boils.
- And you can immediately see, the whole Celsius scale was
- made based on the freezing point and the
- boiling point of water.
- So look at this and you say, if I have something that's 5
- degrees and I have another thing that's 10 degrees, when
- you look at the Celsius scale, you're like, oh, maybe the 10
- degree thing it has twice as much energy as
- the 5 degree thing.
- It has twice the temperature.
- But when you look at it from the absolute distance to zero.
- Let me see if I can draw this.
- So the 10 degree is all the way over here and the 5 degree
- is almost as far.
- So the 10 degrees Celsius is only a slight increment over 5
- degrees Celsius, if you were to divide the two.
- It's not twice as hot.
- And that's why they came up with the Kelvin scale.
- Because in the Kelvin scale, absolute zero is defined as 0.
- So this right here is zero degrees Kelvin.
- And so zero degrees Kelvin is absolute zero.
- So what is zero degrees Celsius?
- And the increments are the same.
- One degree change in Celsius is one
- degree change in Kelvin.
- So at least they keep it, it's just a shift.
- So this is going to be plus 273 degrees Kelvin.
- And then 5 degrees would be plus 278; ' 10 degrees would
- be plus 283 Kelvin.
- And then you see that 5 and 10 degrees really aren't that
- different from each other.
- But in general, if you want to convert from Celsius to Kelvin
- you just add 273 degrees.
- So 30 degrees Celsius is what?
- Well, this 5 and 10 I drew too close to 100.
- But let's say it's sitting here.
- It would be 303 degrees Kelvin.
- All right, so now for our temperature, that's what we
- were worried about.
- We wanted to put in the temperature there.
- So now we can put in our 303 degrees Kelvin.
- Now we have to figure out what constant to use here.
- And I've written a couple of down here.
- Remember, we're dealing with atmospheres and liters.
- So I wrote down a couple of versions of R right here.
- Let's see we're dealing with atmospheres and liters.
- And in the denominator we're always dealing with mole and
- Kelvin no matter what.
- So those are always going to be there.
- So we should use this proportionality constant.
- R is equal to 0.082 liter atmospheres per mole Kelvin.
- Let me write that down.
- So let me rewrite our whole equation actually.
- So I have 2 atmospheres times 2 liters is equal to n times,
- I have a bad memory, 0.082 liter atmospheres per mole
- Kelvin, times 303 degrees Kelvin.
- So let's see what we can do.
- Let's see if all of the units work out.
- So we can always, when you do dimensional analysis, you can
- treat units like numbers.
- So if you divide both sides of this equation by atmospheres,
- the atmospheres cancel out.
- Divide both sides of this equation by liters, liters
- cancel out.
- You have a Kelvin in the numerator, Kelvin in the
- denominator, that cancels out.
- And so we have 2 times 2 is equal to n
- times 0.082 times 303.
- And then we have just a per mole and a 1 over the mole.
- So to solve for n, or the number of moles, what we do is
- we divide both sides of this equation by all of this stuff.
- So we get 2 times 2 is 4.
- 4 divided by 0.082 divided by 303.
- I'm just taking this and putting it on the left-hand
- side, dividing both sides by it.
- And when you divide by a per mole, if you put a 1 over a
- mole here, that's the same thing as
- multiplying by a mole.
- So it's good, the units all worked out.
- We're getting n in terms of moles.
- And so we just have to get the calculator out and figure out
- how many moles we're dealing with.
- So we have 4 divided by 0.082 divided by
- 303 is equal to 0.16.
- If we wanted to go more digits, .161,
- but we'll just round.
- So this is equal to 0.16 moles of H2.
- I am telling you actually here, the exact number of
- hydrogen molecules.
- But if you wanted a number, you'd just multiply this times
- 6.02 times 10 to the 23 and then you would have a number
- in kind of the traditional sense.
- And of course, if you wanted to know what is the mass of
- the hydrogen you have. You'd say, OK well one mole of H2
- has a mass of what?
- The mass of one hydrogen is one atomic mass unit.
- The mass of two hydrogen when it's in its molecular form, is
- two atomic mass units.
- So a mole of it is going to be 2 grams.
- So in this case, we have 0.16 moles.
- So if we wanted to convert that to grams, this in the
- case of these hydrogen gas molecules would be 0.32 grams.
- And I just multiplied it by 2 because each mole is 2 grams.
- Anyway, I hope you found that useful.
- I'm going to do a bunch more of these problems.
- Because I think the math is pretty straightforward here.
- The thing that always makes it daunting, I think, is the
- units and making sure you're using the right units.
- What is they are using meters cubed instead of liters, or
- kilopascals instead of atmospheres.
- So I'll try to do a bunch of examples where we use all the
- different units and you're able to pick our constants
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